1. Field of the Invention
The present invention relates to a permanent magnet motor, and more particularly to an electric motor having a permanent magnet and an armature facing the permanent magnet with a gap therebetween, capable of reducing cogging torque.
2. Description of the Prior Art
A conventionally implemented permanent magnet motor, wherein a cogging torque is reduced by providing auxiliary grooves is disclosed in the Japanese Patent Publication No. 63048147.
According to such permanent magnet motor, however, the cogging torque cannot be reduced sufficiently, because a normal magnetizing yoke is used, so that the magnetic wave form thereof is not suitable.
Minimization of cogging torque generated in the grooves in the iron core
The torque generated in the general electromagnetic machine system can be expressed by Formula 1 under the condition of constant magnetic flux according to the principle of virtual work.                     T        =                  -                                    ∂                              W                m                                                    ∂              θ                                                          (        1        )            
Here, Wm denotes a total magnetic energy, and xcex8 denotes a rotation angle.
The cogging torque will now be considered. The magnetic energy Wm due to the permanent magnet is stored in the magnet, the iron core and the air gap portion. The magnetic energy in the magnet is almost constant, and the energy in the iron core is very small because the iron core has a high permeability. Accordingly, a cogging torque Tc can be expressed by Formula 2 by the angular differentiation of only a magnetic energy Wg in the air gap portion.                               T          c                =                  -                                    ∂                              W                g                                                    ∂              θ                                                          (        2        )            
In order to simplify, it is assumed that the iron core is rotated, and the magnetic energy is stored in a cylindrical air gap portion entirely, and that a magnetic energy is Wg (xcex8) when the relative angle of the stator and the rotor is xcex8. The Wg (xcex8) can be expressed by Formula 3 by integration by rotation at the air gap portion.                                           W            g                    ⁡                      (            θ            )                          =                                                            l                g                            ⁢                              L                S                            ⁢                              r                g                                                    2              ⁢                              xe2x80x83                            ⁢                              μ                0                                              ⁢                                    ∮              C                        ⁢                                                            B                  g                  2                                ⁡                                  (                                      θ                    +                    γ                                    )                                            ⁢                              ⅆ                γ                                                                        (        3        )            
Here, lg denotes an air gap length, Ls denotes an effective thickness of iron core, xcexc0 denotes a vacuum permeability, rg denotes a mean radius of air gap portion, and Bg (xcex8+xcex3) denotes a distribution of the magnetic flux density in the air gap with respect to an angle xcex3 in the iron core rotated by an angle xcex8.
In a smoothed iron core 1 having no winding grooves as shown in FIG. 1, no cogging torque due to the rotation is generated because there are no winding grooves. Accordingly, the magnetic energy Wg (xcex8) in the Formula 3 is constant having no relation to the rotation angle (xcex8). On the contrary thereto, it is considered that if the winding grooves exist, Bg ("xgr") or Bg2 ("xgr") lacks substantially at the angle of xcex3, so that the cogging torque is generated. Here, "xgr"=xcex8+xcex3. The Wg can be expressed by Formulas 4-6, if the lacked magnetic energy due to the winding grooves is xcex4 Wg.                               δ          ⁢                      xe2x80x83                    ⁢                      W            g                          =                              ∑                          k              =              1                        3                    ⁢                                    w              g                        ⁡                          (                              θ                +                                  γ                  k                                            )                                                          (        5        )                                                      w            g                    ⁡                      (                          θ              ,                              γ                k                                      )                          =                                                            l                g                            ⁢                              L                S                                                    2              ⁢                              xe2x80x83                            ⁢                              μ                o                                              ⁢                      k            sk                    ⁢                                    B              g              2                        ⁡                          (                              θ                +                                  γ                  k                                            )                                                          (        6        )            
Here, Wg denotes a magnetic energy in the air gap portion of the smoothed iron core, s denotes a number of grooves, xcex3k denotes an angle of a No. k winding groove, ksk denotes a coefficient determined by a figure of the No. k winding groove, and Bg (xcex8+xcex3k) is a magnetic flux density in the air gap at a position of No. k groove.
By putting the Formulas 4 to 5 in the Formula 2, the cogging torque can be expressed by Formula 7.                               T          c                =                                            ∂                              (                                  δ                  ⁢                                      xe2x80x83                                    ⁢                                      W                    g                                                  )                                                    ∂              θ                                =                                                                      l                  g                                ⁢                                  L                  S                                                            2                ⁢                                  xe2x80x83                                ⁢                                  μ                  o                                                      ⁢                          ∂                              ∂                θ                                      ⁢                          (                                                ∑                                      k                    =                    1                                    s                                ⁢                                                      k                    sk                                    ⁢                                                            B                      g                      2                                        ⁡                                          (                                              θ                        +                                                  γ                          k                                                                    )                                                                                  )                                                          (        7        )            
The right side of the Formula 7 is the sum of magnetic energy portions lost by the winding grooves. It can be said that it is similar to the function of the hole in the semiconductor engineering. Specifically, it can be said that the cogging torque is generated by the reduction of the magnetic energy due to the grooves. Accordingly, a manner for reducing the cogging torque is now studied under the point of view as follows.
FIG. 2 shows results of the distribution of the magnetic flux density in the air gap measured by providing a hole element on the surface of the iron core and rotating the iron core, in order to know a figure of Bg ("xgr"). The analysis is proceeded on the assumption that a figure of the distribution of the magnetic flux density in the air gap is shown in FIG. 3 with respect to the electrical angle p "xgr". xcex2 denotes a ratio of an inclined portion. It is supposed that the magnetic flux density is varied as a figure of a fourth part of a sine wave in a section corresponding to xcex2 shown in Formula 8.
(0 less than xcex2xe2x89xa61)xe2x80x83xe2x80x83(8)
The Bg ("xgr") can be expressed by Formula 9.                                           B            g                    ⁡                      (            ξ            )                          ⁢                  {                                                                      =                                                                                                              -                          1                                                ⁢                                                  xe2x80x83                                                ⁢                        for                                            ⁢                                              xe2x80x83                                            -                                                                        π                          2                                                ⁢                        p                        ⁢                                                  xe2x80x83                                                ⁢                        ξ                                                               less than                                           -                                                                        β                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                2                                                                                                                                                                  =                                                                                    sin                        ⁢                                                  xe2x80x83                                                ⁢                                                                              p                            ⁢                                                          xe2x80x83                                                        ⁢                            ξ                                                    β                                                ⁢                                                  xe2x80x83                                                ⁢                        for                                            ⁢                                              xe2x80x83                                            -                                                                        β                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                2                                                              ≤                                          p                      ⁢                                              xe2x80x83                                            ⁢                      ξ                                        ≤                                                                  β                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            2                                                                                                                                            =                                                            1                      ⁢                                              xe2x80x83                                            ⁢                      for                      ⁢                                              xe2x80x83                                            ⁢                                                                        β                          ⁢                                                      xe2x80x83                                                    ⁢                          π                                                2                                                               less than                                           p                      ⁢                                              xe2x80x83                                            ⁢                      ξ                                        ≤                                          π                      2                                                                                                                              (        9        )            
The Formula 9 can be expressed by Fourier series in the form of Formula 10 consisting of terms of odd number order.                                           B            g                    ⁡                      (            Ϛ            )                          =                              ∑                          n              =              o                        ∞                    ⁢                                    b                                                2                  ⁢                  n                                -                1                                      ⁢                          sin              ⁡                              (                                                      (                                                                  2                        ⁢                        n                                            -                      1                                        )                                    ⁢                  p                  ⁢                                      xe2x80x83                                    ⁢                  ξ                                )                                                                        (        10        )            
The coefficient can be expressed by Formula 11 in case of xcex2=0 and by Formula 12 in case of 0 less than b less than 1.                               b                                    2              ⁢              n                        -            1                          =                  4                                    (                                                2                  ⁢                  n                                -                1                            )                        ⁢            π                                              (        11        )                                          b                                    2              ⁢              n                        -            1                          =                              4                                          (                                                      2                    ⁢                    n                                    -                  1                                )                            ⁢                              π                ⁡                                  (                                                                                                              β                          2                                                ⁡                                                  (                                                                                    2                              ⁢                              n                                                        -                            1                                                    )                                                                    2                                        -                    1                                    )                                                              ⁢          cos          ⁢                      xe2x80x83                    ⁢                                                    (                                                      2                    ⁢                    n                                    -                  1                                )                            ⁢              β              ⁢                              xe2x80x83                            ⁢              π                        2                                              (        12        )            
In case of xcex2=1, only the fundamental wave is presented.
Bg2 ("xgr") can be expressed by Formula 13 which is a even function consisting of terms of even number order.                                           B            g            2                    ⁡                      (            ξ            )                          =                              a            o                    +                                    ∑                              n                =                1                            ∞                        ⁢                                          a                                  2                  ⁢                  n                                            ⁢              cos              ⁢                              xe2x80x83                            ⁢              2              ⁢              np              ⁢                              xe2x80x83                            ⁢              ξ                                                          (        13        )            
FIG. 4 shows the change of each harmonic coefficient a2n of Bg2 with respect to xcex2. When xcex2 is zero, it becomes a square wave, and when xcex2 is 1, it becomes a pure sine wave. The second order component corresponds to the fundamental wave, and becomes larger in value when the order number is smaller in value. The maximum value thereof exists in the middle portion of the change of xcex2.
By putting Formula 13 in Formula 7, Formula 14 can be obtained.                                                                         T                c                            =                            ⁢                                                                                          l                      g                                        ⁢                                          L                      S                                                                            2                    ⁢                                          xe2x80x83                                        ⁢                                          μ                      o                                                                      ⁢                                                      ∑                                          n                      =                      1                                        ∞                                    ⁢                                      [                                                                  ∂                                                  ∂                          θ                                                                    ⁢                                                                        ∑                                                      k                            =                            1                                                    s                                                ⁢                                                                              k                            sk                                                    ⁢                                                      a                                                          2                              ⁢                              n                                                                                ⁢                          cos                          ⁢                                                      xe2x80x83                                                    ⁢                          2                          ⁢                                                      xe2x80x83                                                    ⁢                                                      np                            ⁡                                                          (                                                              θ                                +                                                                  γ                                  k                                                                                            )                                                                                                                                            ]                                                                                                                          =                            ⁢                                                                                          l                      g                                        ⁢                                          L                      S                                                                            μ                    o                                                  ⁢                                                      ∑                                          n                      =                      1                                        ∞                                    ⁢                                      [                                                                  ∑                                                  k                          =                          1                                                s                                            ⁢                                                                        npk                          sk                                                ⁢                                                  a                                                      2                            ⁢                            n                                                                          ⁢                        sin                        ⁢                                                  xe2x80x83                                                ⁢                        2                        ⁢                                                  np                          ⁡                                                      (                                                          θ                              +                                                              γ                                k                                                                                      )                                                                                                                ]                                                                                                          (        14        )            
In order to minimize the cogging torque, it is understood that a sum of components due to the winding grooves should be set to zero as shown in Formula 15 in the most of the harmonics of low order (n=1, 2, 3. . . ) which affect largely on the cogging torque.                                           ∑                          k              =              1                        s                    ⁢                                    npk              sk                        ⁢                          a                              2                ⁢                n                                      ⁢            sin            ⁢                          xe2x80x83                        ⁢            2            ⁢                          np              ⁡                              (                                  θ                  +                                      γ                    k                                                  )                                                    =                  0          ⁢                      xe2x80x83                    ⁢                      (n:  natural  numeral)                                              (        15        )            
This is the principle of minimization of the cogging torque due to the iron core grooves. A manner for reducing the cogging torque with respect to the three-phase permanent magnet motor on the basis of the principle is now considered.
Minimization of the cogging torque in the three-phase winding grooves
A recent conventional small motor of non-lap concentration winding construction will now be studied. The motor generally comprises an annular four-pole permanent magnet, an armature having six magnetic poles, and six winding grooves.
Following conditions must be satisfied for the winding grooves to which three-phase windings can be wound.
(1) A number s of grooves is a multiple number of three.
(2) Three-phase windings having phase difference of 120xc2x0 in electrical angle can be formed.
Here, the component in the Formula 15 is expressed by a plurality of vectors Ank and named as groove vectors.
The groove vectors Ank can be expressed by Formula 16.
Ank=npkskxcex12n∈j2np"xgr"kxe2x89xa1Ano∈j2np"xgr"kxe2x80x83xe2x80x83(16)
FIG. 6A shows vectors in the second harmonic plane of the brushless motor having four-pole permanent magnet and six winding grooves, wherein p=2, s=6 and n=1. FIG. 6B shows vectors in the fourth harmonic plane of the brushless motor having four-pole permanent magnet and six winding grooves, wherein p=2, s=6 and n=2. It is noted from FIG. 6A and FIG. 6B that every three vectors are balanced and the relation of the Formula 15 is certified. However, in case of n=3, all of the vectors A6k are superposed on the same position of 0xc2x0, so that the balance cannot be kept. Accordingly, in this case, a cogging torque is generated by the sixth harmonic.
In general, all groove vectors are balanced and Formula 15 is established so far as anisotropic vectors dividing equally the electrical angle 4np xcfx80 of the harmonic order exist, because No. s angle "xgr"x is 2 xcfx80 (360xc2x0) when xcex8=0. However, the all vectors are superposed in the same direction and not balanced when the distance of the vectors becomes 2i xcfx80 (i is an integer). In such case, Formula 17 is established if s is 3 m (m is a natural numeral).                     m        =                                            2              ⁢              np                                      3              ⁢              i                                ⁢                      xe2x80x83                    ⁢                      (unbalance  condition)                                              (        17        )            
The Formula 17 can be applied to the motor having four-pole permanent magnet and six winding grooves, wherein p=2, m=2, n=3 and i=2. The unbalance condition expressed by the Formula 17 is obtained when p or n is a multiple number of three. In the above motor, the above condition (2) cannot be obtained some times, and the cogging torque is generated also in case that the n is not a multiple number of three.
Accordingly, a combination in which the Formula 17 cannot be established when n is three or a multiple number of three should be selected, in order to reduce the cogging torque.
Table 1 shows a representative example of combinations of the number of the winding grooves and the number of the magnetic poles for minimizing the cogging torque obtained from the above, with respect to the non-lap concentration winding which is excellent windings for the permanent magnet motor. No combination for minimizing the cogging torque exists, because no cogging torque due to the sixth harmonic is generated when the groove number is not more than six. A combination of two grooves and three magnetic poles or three grooves and four magnetic poles used conventionally cannot be said as a combination for minimizing the cogging torque.
Table 1 shows the judgment of the cogging torque balance of twelfth order which should be considered next of the sixth order, wherein cases of the magnetic pole numbers 20 and 28 with the groove number 24 are X (bad), and cases of the magnetic pole numbers 22 and 26 with the groove number 24 are O (good).
FIG. 7A shows relations of vectors in the sixth harmonic with respect to a motor having twelve winding grooves and ten poles (12?S/10-P), and FIG. 7B shows that with respect to a motor having nine winding grooves and eight poles (9-S/8-P).
In the motor shown in FIG. 7A, vectors are concentrated in two vectors deviated by 180xc2x0 from each other and balanced, whereas in the motor shown in FIG. 7B, vectors are concentrated in three vectors deviated by 120xc2x0 from one another and balanced.
Combinations of a small groove number and a small magnetic pole number, which are not included in the Table 1 are studied. In this case, the cogging torque is generated because all groove vectors in the sixth harmonic plane are aligned on a line of zero phase. Accordingly, it is effective to provide auxiliary grooves in positions at which the groove vectors are balanced.
It is considered that (a) the auxiliary grooves are provided in opposite phase positions, and (b) the auxiliary grooves are provided in positions deviated by 120xc2x0 from one another in consideration of FIG. 7A and FIG. 7B. In the case (a), Formula 18 is established if an angle between the winding groove and the auxiliary groove is "xgr".                     ζ        =                              ±                          xe2x80x83                        ⁢                                                            (                                                            2                      ⁢                      i                                        +                    1                                    )                                ⁢                π                                            6                ⁢                p                                              ⁢                      xe2x80x83                    ⁢                      (i:  integer)                                              (        18        )            
Here, i is an integer.
Similarly, in the case (b), Formula 19 is established.                     ζ        =                  ±                      xe2x80x83                    ⁢                                                    (                                  i                  +                                      1                    /                    3                                                  )                            ⁢              π                                      3              ⁢              p                                                          (        19        )            
Representative examples of the auxiliary groove position xcex6 according to the Formulas 18 and 19 are shown in Table 2.
In this case according to the Formula 19, it is necessary to provide the auxiliary grooves in both (+) and (xe2x88x92) angular positions, in order to balance the three vectors.
The invention disclosed in the Japanese Patent Publication No. 63048147 corresponds to the above auxiliary groove system, however, any effect of the auxiliary grooves is not recognized in a state that xcex2, the inclined portion rate, is less than 20%. The effect of the auxiliary grooves can be recognized in a state that xcex2 is more than about 30%. In other words, a large effect can be obtained in a range that a width of an area having a value more than 90% of the peak value of the air gap magnetic flux density is less than 80% of a width of a pole pitch xcfx80. It is difficult to magnetize in practice in the state that the inclined portion rate xcex2 is zero. It is supposed that the inclined portion rate xcex2 is about 20% in case that a conventional magnetizing yoke is used. As a result, it is recognized that the clogging torque is not reduced in the Japanese Patent Publication No. 63048147.
The present invention can be obtained based on the above considerations.
An object of the present invention is to provide a permanent magnet and an armature having a plurality of winding grooves facing the permanent magnet with an air gap therebetween, the armature having a plurality of auxiliary grooves on a magnetic pole surface thereof, wherein a composite of two vectors of cogging torque generated by the auxiliary grooves is deviated by 180xc2x0 from a vector of cogging torque generated by the winding grooves in the sixth harmonic plane, and wherein a width of Portion of an air gap magnetic flux density waveform that includes a value that is more than 90% of a peak value of an air gap magnetic flux density is less than 80% of a width of a pole pitch xcfx80 portion of the air gap magnetic flux density waveform.
Another object of the present invention is to provide a permanent magnet and an armature having a plurality of winding grooves facing the permanent magnet with an air gap therebetween, the armature having a plurality of auxiliary grooves on a magnetic pole surface thereof, wherein each vector of cogging torque generated by each of the plurality of auxiliary grooves is deviated by 180xc2x0 from a vector of cogging torque generated by the winding grooves in the sixth harmonic plane, respectively, and wherein a width of portion of an air gap magnetic flux density waveform that includes a value that is more than 90% of a peak value of the air gap magnetic flux density waveform is less than 80% of a width of a pole pitch xcfx80 portion of the air gap magnetic flux density waveform.
A further object of the present invention is to provide a permanent magnet and an armature having a plurality of winding grooves facing the permanent magnet with an air gap therebetween, the armature having a plurality of auxiliary grooves on a magnetic pole surface thereof, wherein a composite two vectors of cogging torque generated by the auxiliary grooves is deviated by 120xc2x0 from a vector of cogging torque generated by the winding grooves in the sixth harmonic plane, and wherein a width of portion of an air gap magnetic flux density waveform that includes a value that is more than 90% of a peak value of the air gap magnetic flux density waveform is less than 86% of a width of a pole pitch xcfx80 portion of the air gap magnetic flux density waveform.
Yet further object of the present invention is to provide a permanent magnet and an armature having a plurality of winding grooves facing the permanent magnet with an air gap therebetween, the armature having a plurality of auxiliary grooves on a magnetic pole surface thereof, wherein each vector of cogging torque generated by each of the plurality of auxiliary grooves is deviated by 120xc2x0 from a vector of cogging torque generated by the winding grooves in the sixth harmonic plane, respectively, and wherein a width of portion of an air gap magnetic flux density waveform a value more than 90% of a peak value of is less than 86% of a width of a pole pitch xcfx80 portion of the air gap magnetic flux density waveform.
The permanent magnet is an inner rotor made of a pole-magnetized magnet.
The forgoing and other objects, features, and advantages of the present invention will become apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings.